1,026 research outputs found
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
Control landscapes for two-level open quantum systems
A quantum control landscape is defined as the physical objective as a
function of the control variables. In this paper the control landscapes for
two-level open quantum systems, whose evolution is described by general
completely positive trace preserving maps (i.e., Kraus maps), are investigated
in details. The objective function, which is the expectation value of a target
system operator, is defined on the Stiefel manifold representing the space of
Kraus maps. Three practically important properties of the objective function
are found: (a) the absence of local maxima or minima (i.e., false traps); (b)
the existence of multi-dimensional sub-manifolds of optimal solutions
corresponding to the global maximum and minimum; and (c) the connectivity of
each level set. All of the critical values and their associated critical
sub-manifolds are explicitly found for any initial system state. Away from the
absolute extrema there are no local maxima or minima, and only saddles may
exist, whose number and the explicit structure of the corresponding critical
sub-manifolds are determined by the initial system state. There are no saddles
for pure initial states, one saddle for a completely mixed initial state, and
two saddles for other initial states. In general, the landscape analysis of
critical points and optimal manifolds is relevant to the problem of explaining
the relative ease of obtaining good optimal control outcomes in the laboratory,
even in the presence of the environment.Comment: Minor editing and some references adde
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
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